Uncertainty in Hard, Soft and Hard-Soft

Modeling

Uncertainty in Calculated Model

Parameters using Hard- Modeling Method

The very rigid constraints of a chemical model form a framework within which the fit is confined and which results in a robust analysis, in model-free analysis, this framework is dramatically wider and looser and these methods suffer gradually from a sever lack of robustness. It must be remembered, however, that the choice of the wrong model necessarily results in the rung analysis and wrong resulting parameters.

Model Based Analyses

Complex Formation Equilibrium

M + L ML [M] [L][ML ]

Kf =[ ]

CL = [L] + [ML]CM = [M] + [ML ]

CM = [M] + KF [M] [L]

CL = [L] + KF [M] [L]

Data.m

Spectrophotometric monitoring of complex

formation titration

200 250 300 3500

500

1000

1500

2000

2500

Wavelength

Molar A

bsorptivity

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10-3

Mole Ratio, CM/CL

[spe

cies

]

LML

200 250 300 350-0.5

0

0.5

1

1.5

2

Wavelength

Abs

orba

nce

Calculation of Model ParameterThe task of model-based data fitting for a given matrix A, is to determine the best parameters defining matrix C, as well as the best pure responses collected in matrix E.

A = C E + R

A C E R= +

The quality of the fit is represented by the matrix of residuals. Assuming white noise, the sum of the squares, ssq, of all elements ri,j is statistically the best measure to be minimized ssq = ΣΣ r2 I,j

R = A – C E = A – C C+ A = f( A, model, K)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

5

10

15

20

25

30

log beta

ssq

Calculation of Model Parameter

How we can calculate the precision of model parameter?

3.4 3.42 3.44 3.46 3.48 3.5 3.52 3.54 3.56 3.58 3.6

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

log beta

ssq

Repeatation

3.496 3.497 3.498 3.499 3.5 3.501 3.502 3.503 3.504 3.5050

2

4

6

8

10

12

14

Calculated K

Freq

uenc

y

Distribution of 50 calculated KDistribution of Fitted Model Parameters

log (Kf) (mean) = 3.5004 Standard Deviation of log (Kf)= 0.0021

Main_ML_S.m

Search for K in a certain range

?Based on repeatation procedure, calculate the standard deviation of fitted parameter in different level of noise

Error Propagation

y = f (x)

var (y) = (df/ dx)2 var (x)

y = f (x1, x2)var (y) = (df/dx1)2 var (x1) + (df/dx2)2 var(x2) + (df/d(x1) d(f)/d(x2) 2cov(x1, x2)

Var(x) =

var(x1), cov(x1, x2), … , cov(x1,xn) cov(x2, x1), var(x1), … , cov(x2, xn)

… … …

cov(xn, x1), cov(xn, x2), … , var(xn)

JT= [ df/dx1, df/dx2, …, df/dxn]

y = f (x1, x2, x3, …)var (y) = JT [Var (x)] J

General Error Propagation

R = A – C E = A – C C+ A = f( A, model, p)A = C E + R

Var(p) =

var(p1), cov(p1, p2), … , cov(p1,pn) cov(p2, p1), var(p2), … , cov(p2, pn)

… … …

cov(pn, p1), cov(pn, p2), … , var(pn)

var (R) =JT [Var(p)] JVar(p) =(JT J)-1 var (R)

var (R) = (Ri,j)2/(nm-np) = ssq/df

Uncertainty of fitted model parameters

R(p1+p1) – R(p1-p1)J1= dR/dp1= 2p1

J1 …J2 JnJ =

…

JnJ2J1

J =

Jacobian Matrix

JT J =

J1TJ1 J1

TJ2 … J1TJn

J2TJ1 J2

TJ2 … J2TJn ………

JnTJ1 Jn

TJ2 … JnTJn

Hessian Matrix

The inverted Hessian matrix H-1, is the variance-covariance matrix of the fitted parameters. The diagonal elements contain information on the parameter variances and the off-diagonal elements the covariances.

Newton-Gauss-Levenberg-Marquardt Algorithmguess parameters, p=pstart initial value for mp

Calculate residuals, r(p) and sum of squares, ssq

ssqold< = > ssq

Calculate Jacobian, J

Calculate shift vector p, and p = p + p

End, display results

=

>

mp=0

mp=0

<

mp ×10 mp / 3

yes

no

Main_ML.m

NGLM algorithm for Fitting

?Use Main_ML m-file for fitting the three parameters (K, CM and CL) with different initial estimates

?Check the uncertainty calculated for K when the initial concentrations are fixed or fitted

Correlation between Fitted Parameters

When two parameters are fitted, is there any relation between calculated parameters?

Is there any relation between the estimated uncertainties on K and C0?

?????????????????????????????????????????

KC0

Main_ML_corr.m

Correlation between fitted parameters

?What are the relations between the shapes and values of Jacobian with variance and covariance of parameters?

?Using the J matrix and calculate the corelation between parameters

Propagation of Uncertainty from Initial Concentration to Equilibrium Constant

K = f(residual, C0)

var(K) = (df/d(residual))2var(residual) +

(dK/dC0)2 var (C0)

(dK/dC0)2 Sensitivity of K to C0

Kopt(C0+C0) – Kopt(C0-C0)dK/dC0 =

2C0

Main_ML_C0

Propagation of uncertainty from C) to K

?What is the effect of noise on measured signal in uncertainty of K due to C0?

?Modify the Main_ML_cO m-file for considering the uncertainty in C0

M and C0L